Mori dream fibers and the geometric generic fiber
Abstract
We construct a smooth projective family of rational surfaces over 𝔾m,ℤ. The Mori dream property of a fiber is determined by the torsion of the normal bundle of an anticanonical cycle. Over ℂ, the locus of Mori dream fibers is Zariski dense. For every prime p, every geometric fiber over a closed point of the reduction modulo p is a Mori dream surface, whereas the geometric generic fiber is not a Mori dream space. In either setting, no restriction to a nonempty open subset is a Mori dream morphism. We also prove that, over any algebraically closed field, a projective fibration becomes a Mori dream morphism after shrinking the base whenever the set of points with Mori dream fibers is not contained in a countable union of proper closed subsets.
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BibTeX
@misc{lee2026mori,
title = {{Mori dream fibers and the geometric generic fiber}},
author = {Dae-Won Lee and Masaru Nagaoka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17103},
eprint = {2609.17103},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17103},
}
RIS
TY - UNPB
TI - Mori dream fibers and the geometric generic fiber
AU - Dae-Won Lee
AU - Masaru Nagaoka
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17103
UR - https://arxiv.org/abs/2609.17103
AB - We construct a smooth projective family of rational surfaces over 𝔾_m,ℤ. The Mori dream property of a fiber is determined by the torsion of the normal bundle of an anticanonical cycle. Over ℂ, the locus of Mori dream fibers is Zariski dense. For every prime p, every geometric fiber over a closed point of the reduction modulo p is a Mori dream surface, whereas the geometric generic fiber is not a Mori dream space. In either setting, no restriction to a nonempty open subset is a Mori dream morphism. We also prove that, over any algebraically closed field, a projective fibration becomes a Mori dream morphism after shrinking the base whenever the set of points with Mori dream fibers is not contained in a countable union of proper closed subsets.
ER -